Completely bounded maps between sets of Banach space operators.

*(English)*Zbl 0747.46015Let \(X\) and \(Y\) be Banach spaces, let \(1\leq p<\infty\), and let \(S\) be a subspace of \(B(X,Y)\), the space of all bounded operators from \(X\) into \(Y\). A norm on \(M_ n(S)\), the space of all \(n\times n\) matrices with entries in \(S\), is obtained by \(\|(a_{ij})\|_ p=\sup(\sum_ i\|\sum_ ja_{ij}x_ j\|^ p)^{1/p}\), the supremum being extended over all \(n\)-tuples \((x_ j)_{j\leq n}\) in \(X\) such that \(\sum_ j\| x_ j\|^ p\leq 1\).

If \(X_ 1\), \(Y_ 1\) are further Banach spaces, then a linear map \(u:S\to B(X_ 1,Y_ 1)\) is called \(p\)-completely bounded \((p.c.b)\) if there is a constant \(C\) such that, for any \(n\) and any \((a_{ij})\in M_ n(S)\), \(\|(u(a_{ij}))\|_ p\leq C\|(a_{ij})\|_ p\); let \(\| u\|_{pcb}\) be the least of all such \(C\)’s. In a Hilbert space situation, the case \(p=2\) corresponds to the usual definition of complete boundedness known from the theory of \(C^*\)-algebras.

Guided by a formal analogy of the definition of \(\|\cdot\|_{2cb}\) with a characterization of \(L_ 2\)-factorability of Banach space operators due to J. Lindenstrauss and A. Pełczyński [Stud. Math. 29, 275–326 (1968; Zbl 0183.40501)], the author starts by giving a rather direct proof of the following generalization of G. Wittstock’s factorization theorem [J. Funct. Anal. 40, 127–150 (1981; Zbl 0495.46005)]. Let, in the above situation, \(X=Y=H\) be a Hilbert space, and let \(u:S\to B(X_ 1,Y_ 1)\) be 2.c.b. Then there is a Hilbert space \(\hat H\), along with a representation \(\pi:B(H)\to B(H_ 1)\) and operators \(V_ 1:X_ 1\to\hat H\) and \(V_ 2:\hat H\to Y_ 1\), such that \(u(a)=V_ 2\pi(a)V_ 1\) for each \(a\in S\) and \(\| V_ 1\|\cdot\| V_ 2\|\leq\| u\|_{2cb}\), the latter inequality being then automatically an equality.

Using ultraproducts of vector valued \(L_ p\) spaces, it is possible to extend this to the general Banach space situation \(S\subset B(X,Y)\), \(1\leq p<\infty\), but the statement becomes more complicated. Nevertheless, if \(X=Y_ 1=C\), then an operator \(u\) from \(S=Y\) to \(B(X_ 1,Y_ 1)=X^*_ 1\) is p.c.b. if and only it is absolutely \(p^*\)- summing \((p^*=p/(p-1))\), and the above extension is just Pietsch’s factorization theorem for such operators. There are numerous further special results, depending on \(X,Y\) having particular properties, such as type, cotype,...

If \(X_ 1\), \(Y_ 1\) are further Banach spaces, then a linear map \(u:S\to B(X_ 1,Y_ 1)\) is called \(p\)-completely bounded \((p.c.b)\) if there is a constant \(C\) such that, for any \(n\) and any \((a_{ij})\in M_ n(S)\), \(\|(u(a_{ij}))\|_ p\leq C\|(a_{ij})\|_ p\); let \(\| u\|_{pcb}\) be the least of all such \(C\)’s. In a Hilbert space situation, the case \(p=2\) corresponds to the usual definition of complete boundedness known from the theory of \(C^*\)-algebras.

Guided by a formal analogy of the definition of \(\|\cdot\|_{2cb}\) with a characterization of \(L_ 2\)-factorability of Banach space operators due to J. Lindenstrauss and A. Pełczyński [Stud. Math. 29, 275–326 (1968; Zbl 0183.40501)], the author starts by giving a rather direct proof of the following generalization of G. Wittstock’s factorization theorem [J. Funct. Anal. 40, 127–150 (1981; Zbl 0495.46005)]. Let, in the above situation, \(X=Y=H\) be a Hilbert space, and let \(u:S\to B(X_ 1,Y_ 1)\) be 2.c.b. Then there is a Hilbert space \(\hat H\), along with a representation \(\pi:B(H)\to B(H_ 1)\) and operators \(V_ 1:X_ 1\to\hat H\) and \(V_ 2:\hat H\to Y_ 1\), such that \(u(a)=V_ 2\pi(a)V_ 1\) for each \(a\in S\) and \(\| V_ 1\|\cdot\| V_ 2\|\leq\| u\|_{2cb}\), the latter inequality being then automatically an equality.

Using ultraproducts of vector valued \(L_ p\) spaces, it is possible to extend this to the general Banach space situation \(S\subset B(X,Y)\), \(1\leq p<\infty\), but the statement becomes more complicated. Nevertheless, if \(X=Y_ 1=C\), then an operator \(u\) from \(S=Y\) to \(B(X_ 1,Y_ 1)=X^*_ 1\) is p.c.b. if and only it is absolutely \(p^*\)- summing \((p^*=p/(p-1))\), and the above extension is just Pietsch’s factorization theorem for such operators. There are numerous further special results, depending on \(X,Y\) having particular properties, such as type, cotype,...

Reviewer: Hans Jarchow (Zürich)

##### MSC:

46B28 | Spaces of operators; tensor products; approximation properties |

46B08 | Ultraproduct techniques in Banach space theory |

46B07 | Local theory of Banach spaces |

46A32 | Spaces of linear operators; topological tensor products; approximation properties |

46L10 | General theory of von Neumann algebras |